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Unformatted text preview: Matching Markets: Lecture 7 Georgios Katsenos Department of Economics, University of Hannover December 03, 2009 Housing Market Housing Market: Agents enter the market already owning one house. They can improve upon their property via exchange.- It defers from the house allocation problem in assuming the presence of endowments.- It is essentially a pure-exchange economy with discrete resources (1 unit of each commodity).- Problem formulated by Shapley and Scarf (1974). The algorithm presented is attributed to David Gale. Housing Market The Housing Market:- Connection to the house allocation problem:- Assign the houses randomly, by lottery.- Allow the agents to exchange houses. How does the expected solution relate to the outcome of the serial dictatorship with randomly assigned priorities?- Important application:- Exchange of live kidney donors. Formal Model Model: The setting is described by { ( a i ,h i ,P i ) } n i =1 , where- A = { a 1 ,...,a n } is the set of agents.- H = { h 1 ,...,h n } is the set of houses.- The house h i is initially owned by the agent a i .- P i is agent a i s preference ordering of the houses. We assume that the preferences are strict. Matchings Matching: A matching in { ( a i ,h i ,P i ) } n i =1 is a 1-1 function : A H ; that is, if a 6 = a , then ( a ) 6 = ( a ). Pareto Efficient Matching: Defined exactly as in the housing allocation market: There is no matching in which all agents are weakly better-off and at least one agent is strictly better-off. Individually Rational Matching: A matching such that ( a ) a h a , for all a A . No agent gets a worse house than the one he had initially. Mechanisms Mechanism: A mechanism associates each market { ( a i ,h i ,P i ) } n i =1 with a matching via a revelation game. Pareto Efficient Mechanism: A mechanism that produces Pareto efficient matchings, for any preferences reported. Individually Rational Mechanism: A mechanism that produces individually rational matchings, for any preferences reported. Strategy-proof Mechanism: A mechanism in which truth-telling, Q i = P i , is a dominant strategy for all agents. Stability: The Core The Core: Our concept of stability requires that no group of agents will be better-off blocking a matching by exchanging only within the group. Definition: A matching is in the core of the market { ( a i ,h i ,P i ) } n i =1 if there is no coalition of agents B A and a matching such that- For all a B , we have ( a ) { h i } a i B ;- ( a ) a ( a ), for all a B ;- ( a ) a ( a ), for at least one a B . Stability: The Core Remarks:- In many-to-one, two sided matching, stability is equivalent to the core....
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